Estimating Particle Size Using Distributions

Particle size helps determine the appearance of paint, the flavor of cocoa powder, the strength of cement, the properties of die filling powder, the absorption rates of pharmaceuticals, and the appearance of cosmetics, according to Horiba’s guidebook to particle size analysis. Determining particle size enables industries to control the quality of their products, optimize manufacturing efficiency, and achieve compliance in regulated markets.

Spherical equivalents

When trying to assess particle size, researchers routinely measure particle diameter, the assumption being that all dimensions of the particles in question are identical. The many measures that factor into the size of a sphere—its perimeter, projected cross-sectional area, surface area, and volume—can be described unambiguously by diameter. Furthermore, the diameter of a sphere remains constant regardless of the angle of view.

However, things quickly become complicated when particles are not perfectly spherical. Other than a sphere, no regular or irregular shape projects the same cross-section at all angles. That means neither surface area nor volume can be inferred from the cross-section of a nonspherical particle.

One way to describe the size of nonspherical particles is by using multiple horizontal and vertical measures; however, such descriptions are complex. Therefore, it is common for researchers to assume that all particles in a sample are spherical and to report the value as a one-dimensional sphere-equivalent by determining the size of the sphere that could have produced the values obtained by scattered light, settling rate, or other methods.

The need for distributions

Obtaining the spherical equivalent, however, is less than ideal in certain situations. For irregular shapes with large aspect ratios, such as rods and fibers, size can differ significantly depending on which dimension you measure. To deal with this issue, samples are often represented as distributions of sizes. In its white paper on the topic, Malvern Instruments Limited (Worcestershire, UK) outlines several types of particle size distributions, which include:

Number-weighted distributions: counting techniques such as image analysis can enable each particle to be weighted irrespective of its size.

Volume-weighted distributions: static lightscattering techniques such as laser diffraction yield volume-weighted distributions, where the relative contribution of a particle is proportional to its size.

Intensity-weighted distributions: dynamic lightscattering techniques yield intensity-weighted distributions, where each particle’s contribution to the distribution is related to the intensity of light scattered by the particle.

Depending on which type of distribution is being used to measure particle size in a sample, one might get widely different results; for example, a volume-weighted distribution based on image analysis will not necessarily match a distribution based on laser diffraction.

Conversion of data from one type of distribution to another can be done, but it requires assumptions to be made about the physical properties of the particles.

With a distribution in hand, the next task is reporting results. Depending on the information required, several descriptive statistics may be suitable. The mean, median, and mode all describe the central value of the distribution. Standard deviation and coefficient of variation describe the width of the distribution. Ideally, a central value will be combined with a width measure to give the most complete picture of the particle sizes represented by the distribution.

It’s not all about size

In addition to particle size, certain industries are interested in measuring particle shape, which has been described as “the second variable of the particle characterization equation.” Advanced imaging techniques are providing those industries with crucial information needed to supplement their size distributions with greater details about the shapes of the particles in their samples.

For additional resources on particle sizing, including useful articles and a list of manufacturers, visit www.labmanager.com

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